Rigidity of symmetric frameworks in normed spaces

Abstract

We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of $d$-dimensional normed spaces (including all ${\ell }^p$ spaces with $p\ne 2$). Complete combinatorial characterisations are obtained for half-turn rotation in the ${\ell }^1$ and ${\ell }^{\infty }$-plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of $(2,2,0)$-gain-tight graphs.

42 pages, 11 figures. This version contains corrected proofs and more detailed explanations have been provided for greater clarity